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Conference Papers Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics Year : 2020

Statistical Estimation of the Poincaré constant and Application to Sampling Multimodal Distributions

Abstract

Poincaré inequalities are ubiquitous in probability and analysis and have various applications in statistics (concentration of measure, rate of convergence of Markov chains). The Poincaré constant, for which the inequality is tight, is related to the typical convergence rate of diffusions to their equilibrium measure. In this paper, we show both theoretically and experimentally that, given sufficiently many samples of a measure, we can estimate its Poincaré constant. As a by-product of the estimation of the Poincaré constant, we derive an algorithm that captures a low dimensional representation of the data by finding directions which are difficult to sample. These directions are of crucial importance for sampling or in fields like molecular dynamics, where they are called reaction coordinates. Their knowledge can leverage, with a simple conditioning step, computational bottlenecks by using importance sampling techniques.
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Dates and versions

hal-02327453 , version 1 (22-10-2019)
hal-02327453 , version 2 (25-10-2019)
hal-02327453 , version 3 (21-11-2019)

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Loucas Pillaud-Vivien, Francis Bach, Tony Lelièvre, Alessandro Rudi, Gabriel Stoltz. Statistical Estimation of the Poincaré constant and Application to Sampling Multimodal Distributions. AISTATS 2020 : 23rd International Conference on Artificial Intelligence and Statistics, Jun 2020, Palermo / Virtual, Italy. ⟨hal-02327453v3⟩
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